326_midterm-11-solution

# 326_midterm-11-solution - -2 ∑ i ² ˆ U i ³ 2 = 1 n-2...

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UBC, ECONOMICS 326 - 003 2011 MIDTERM EXAMINATION Suggested solution Question 1 First, since ¯ Y = α + β ¯ X + ¯ U, we have ˆ α = α - ( ˜ β - β ) ¯ X + ¯ U . Next, E α | X 1 , . . . , X n ) = α - ( E ( ˜ β | X 1 , . . . , X n ) - β ) ¯ X + E ( ¯ U | X 1 , . . . , X n ) = α - 0 · ¯ X + 0 . The equality holds because ˜ β is conditionally unbiased and therefore E ( ˜ β | X 1 , . . . , X n ) = β , and because E ( ¯ U | X 1 , . . . , X n ) = E n - 1 n X i =1 U i X 1 , . . . , X n ! = n - 1 n X i =1 E ( U i | X 1 , . . . , X n ) = 0 . Lastly, by the LIE, E α ) = E ( E α | X 1 , . . . , X n ) = E ( α ) = α. Question 2 B. Since the p -value is 0.191 and n = 23 , the t - statistic value t can be found from the t -table according to P ( t 21 > t ) = 0 . 191 / 2 0 . 10 , where t 21 denotes a t -distributed random variable with 21 df’s. From the t -table, t 1 . 323 . A. SE = ˆ β 1 /t 0 . 091 / 1 . 323 0 . 069 . C,D. CI 0 . 95 = ˆ β 1 ± t 21 , 0 . 975 × SE = 0 . 091 ± 2 . 08 × 0 . 069 [ - 0 . 053 , 0 . 235] . Question 3 (a) ˆ β * 1 = i ( X * i - ¯ X * ) Y * i i ( X * i - ¯ X * ) 2 = i ( c 2 X i - c 2 ¯ X ) c 1 Y i i ( c 2 X i - c 2 ¯ X ) 2 = c 1 c 2 i ( X i - ¯ X ) Y i c 2 2 i ( X - ¯ X ) 2 = c 1 c 2 ˆ β 1 . (b) ˆ β * 0 = ¯ Y * - ˆ β * 1 ¯ X * = c 1 ¯ Y - c 1 c 2 ˆ β 1 c 2 ¯ X = c 1 ¯ Y - c 1 ˆ β 1 ¯ X = c 1 ˆ β 0 . (c) First, ˆ U * i = Y * i - ˆ β * 0 - ˆ β * 1 X * i = c 1 Y i - c 1 ˆ β 0 - c 1 c 2 ˆ β 1 c 2 X i =

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Unformatted text preview: -2 ∑ i ² ˆ U * i ³ 2 = 1 n-2 ∑ i ² c 1 ˆ U i ³ 2 = c 2 1 s 2 . 1 (d) For H : β * 1 = 0 , we have T * = ˆ β * 1 / s s 2 * / X i ( X * i-¯ X * ) 2 = c 1 c 2 ˆ β 1 / s c 2 1 s 2 / X i ( c 2 X i-c 2 ¯ X ) 2 = c 1 c 2 ˆ β 1 / s ( c 1 /c 2 ) 2 s 2 / X i ( X i-¯ X ) 2 = ˆ β 1 / s s 2 / X i ( X i-¯ X ) 2 = T. Note that T is the test statistic for testing H : β 1 = 0 . (e) Since T = T * and df’s are the same in both cases, pval = pval * . Thus, rescaling the dependent variable and regressor has no eﬀect on testing for signiﬁcance of the slope parameter. 2...
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